Poroelastic properties of rocks with a comparison of theoretical estimates and typical experimental results

The paper develops theoretical estimates for the parameters that describe the classical theory of poroelasticity for a fluid-saturated porous medium, with a porous elastic skeleton that can exhibit imperfect grain contacts. The results for the poroelastic properties predicted from the modelling are compared with experimental results available in the literature.

www.nature.com/scientificreports/ of the idealized concepts relating to the definitions of the granular assembly, porosity, contact stiffnesses and hydraulic aperture at grain contacts are shown in Fig. 1.
References to these studies and other relevant expositions on contact mechanics can be found in the studies [63][64][65][66][67][68] . Applications with special reference to geomaterials are given in [69][70][71] . The interfaces are considered to be very thin and are assumed to have no influence on the overall porosity φ . Denoting the vector normal to the interface by n , the traction vector acting on the interface is denoted by T , the displacement jump at the interface is denoted by [u] and the pore pressure is denoted by p . We can write the constitutive equations for the interface in the form   (Fig. 1) the porosity is expected to have no influence on the interface stiffnesses k n and k t . In addition, we can consider the stress ⌢ T acting on a grain G i at the interface between the grains G i and G j . We can also relate this stress to the displacement jump [u] . Let the coefficients of proportionality in these relations for the normal and tangential components of the stress vector ⌢ T be denoted by K n and K t , respectively. It is shown 57 that Estimates of the elastic properties are obtained from a self-consistent method that is applicable to particulate aggregates developed in [72][73][74] and summarized in 75 . The following normalized quantities are introduced where R is the radius of the grain. Experimental values for the grain sizes of most rocks is provided by granulometric data. In what follows, we make the assumption that the grains are rigid. In this case, the self-consistent estimate of the shear modulus can be obtained by solving the following cubic equation: Once the normalized shear modulus M is known, the self-consistent estimate of the bulk modulus can be obtained from From the above relationship, we have Therefore, if the effective shear modulus G eff and the effective bulk modulus K eff are known from experiments, one can determine the constant M from (6) and consequently estimate the ratio ρ = k t /k n using (4): i.e.
It should be noted 76,77 that self-consistent schemes are not without flaws and should be judiciously applied when extreme limiting cases are being considered. Figure 2 shows estimates of interface stiffness ratio ρ = k t /k n plotted against the porosity φ . The estimates are obtained from (7), based on the values of bulk and shear moduli listed in Table 1 44 and Table 2 43 . It can be observed that the ratio ρ ranges between 0.3 and 0.6 for sandstones and is approximately equal to 0.1 for limestones and granites-marbles.

Numerical results for deformability behaviour
It is also instructive to plot the ratio ρ as a function of the parameter (G eff /K eff ) = (M/K) . Figure 3 shows the ratio ρ plotted against (M/K) . It turns out that ρ is approximately a linear function of the parameter (M/K).
The interfacial stiffness K n R can be obtained from the definition of M (3), i.e., Using (6), we also obtain the ratio of the interfacial stiffness K n R to the effective bulk modulus as follows: Figure 4 shows the variation in the normalized interfacial stiffness (K n R/K eff ) with porosity φ . It can be seen that this dependency can be conveniently approximated by a linear relationship.
When the grains are rigid, the estimate for the Biot coefficient α can be obtained in the form  (6), we can represent the Biot coefficient as   Tables 1 and 2. The grains are modelled as nearly spherical shapes and the pore space is also modelled as a nearly spherical shape. The results can be influenced by the shape of the pores and this was addressed in a previous study 53 .
In Fig. 6 we indicate the dependency of the Biot coefficient α on the porosity φ . This figure demonstrates that by properly choosing the microstructural parameters M (or K ), ρ and r it is possible to exactly match the elastic constants K eff , G eff and α with the experimental data. It can be observed from Fig. 6 that, on average, for sandstones and limestones the Biot coefficient is equal to 0.75 and for the marbles-granite group the Biot coefficient can be lower and in the range 0.2 to 0.5. This observation is consistent with results obtained recently for the Lac du Bonnet granite recovered from the Canadian Shield 79 .
The self-consistent estimate of Biot modulus N can be obtained from the relationship 57 When the estimates for the Biot modulus N obtained from the relationship (13) are compared with the experimental values of N there is a large discrepancy. To eliminate this discrepancy, the compressibility of the fluid phase must be taken into account, which gives another expression for the Biot modulus  www.nature.com/scientificreports/ where K f is the bulk modulus of the fluid. The interface stiffness K n R was estimated first from Eq. (9). This value is then substituted into (13) or (14) to estimate the Biot modulus N . Thus, a knowledge of the grain size radius is not required. Figure 7 shows experimental values of the Biot modulus N and the corresponding theoretical estimates for N obtained from (14). Theoretical values are obtained by setting the bulk modulus of the fluid K f = 2 GPa , which is typically applicable to water. An acceptable agreement between the theoretical and experimental values is obtained by choosing this constant value of K f .

Estimates for permeability
The self-consistent estimate of permeability k is obtained as where η is the interface permeability, which can be related to the aperture of the contact through the usual parallel plate model 55,[80][81][82][83] Here e is the spacing between two neighboring grains (joint opening) and R is the grain radius. By substituting (16)

into (15) we obtain
The result (17) contains the additional factors that can influence permeability, including the grain size, bulk porosity and a measure of the fraction of open interfaces. If these parameters can be accurately estimated, the result gives an added perspective for estimating permeability. Since the equation for permeability (17) is used as an equality constraint in the optimization problem and the solution to the optimization problem exists, then the proper choice of grain radius and joint opening allows us to match the permeability obtained from theoretical estimates (17) with the experimental data.
It should be noted that the definition of the hydraulic aperture does not account for any fine particles that can be present in sandstone and carbonate rocks. As has been suggested 84 , an average value for the joint opening can be set as e = 0.5 µm and for the radius R = 50 µm . It should be noted that these assigned values are typical of tight sandstones and should not be construed as an "average value". They fulfil the function of normalizing values. However, by using these particular values it is not possible to match experimental values of permeability,   (17), we obtain We can rewrite (19) as an equality constraint in terms of variables R n and e 3 n : i.e.
Here k is the value of permeability measured experimentally. As the objective function to be minimized, we can take the sum of deviations of variables R n and e n from unity i.e., We note that R n and e 3 n must be positive, which constitutes inequality constraints. Solution to this non-linear optimization problem for the rocks presented in Table 1 is shown in Table 3. The solution was obtained with the help of MATLAB function fmincon. It may be observed that R n and e n are close to unity only for Berea sandstone and Boise sandstone. For other sandstones, the radius R can be twice 50 µm and the spacing between neighboring grains can be 5-7 times smaller than the normative value of 0.5 µm . The largest deviations are observed for the granite-marble group. Here, the grain radius R is about 5 to 6 times larger than the normative value of 50 µm , and e is , on average, 20 times smaller than 0.5 µm.

Concluding remarks
The estimation of the poroelasticity properties of rocks can proceed along two avenues; the first involves their direct evaluation from experimental data and the second approach is to utilize the theories developed for multiphasic composites to arrive at estimates for the poroelasticity parameters. Both approaches have their advantages and disadvantages. The purely experimental approach is void of consideration of the micro-mechanical input to the poroelastic parameters but provides the data set that can be used in the engineering calculation of poroelastic responses. The approaches based on multiphasic theories have to rely on idealized theoretical assumptions that lead to parameter estimates but adds a new dimension by introducing the properties of the fabric that compose the poroelastic material. The prudent option is to rely on both approaches and to take advantage of the merits of each approach to provide the validations that will enable the assignment of poroelastic parameter estimates. The results of the paper provide correlations between (1) porosity of a poroelastic solid and the grain-grain contact stiffnesses at the particulate level, (2) the interface stiffness ratio and effective shear to bulk modulus ratio, (3) open interface fraction and porosity, (4) the variation of the Biot coefficient and Biot modulus on porosity and (5) the dependency of permeability on microstructural properties. The research utilizes the material data available in the literature and provides the summary related to (1) to (4) in terms of the basic rock types relevant to sedimentary (sandstones, limestones) and igneous (granite, basalt) rocks. The data presented enables the preliminary identification of poroelastic parameters, which can be complemented by a rigorous program of laboratory tests. In the absence of experimental data for poroelastic properties of a particular rock, the findings of the research can be treated as a data set for preliminary geosciences calculations that requires recourse to the theory of linear poroelasticity. Since the original experimental data does not contain ranges or error estimates, the study can provide only firm values of the parameters. The measurement of the properties of saturated rocks with low permeability is generally a challenging task in the laboratory since it is difficult to ensure that the sample is fully saturated during testing. Such theoretical approaches can have practical applications in geotechnical ; e n = e 0.5 µm r(0.5) 3 10 12 R n − e 3 n = 0 (21) min : S = R n (if R n ≥ 1) + 1 R n (if R n < 1) + e n (if e n ≥ 1) + 1 e n (if e n < 1) www.nature.com/scientificreports/ engineering and rock mechanics. Finally, the paper presents a canonical methodology for including additional parameters in the development of concepts for examining poroelastic parameters. In efforts of this nature, the microstructural parameters are introduced within the framework of a plausible concept. The progress of the modelling will require novel experimental methodologies, in the area of geomechanics and material science for accurately estimating the parameters arising from the developments. www.nature.com/scientificreports/ Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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